33rd IMO 1992 shortlisted problems

1.  m is a positive integer. If there are two coprime integers a, b such that a divides m + b2 and b divides m + a2, show that we can also find two such coprime integers with the additional restriction that a and b are positive and their sum does not exceed m + 1.
2.  a and b are positive reals. Let X be the set of non-negative reals. Show that there is a unique function f:X → X such that f( f(x) ) = b(a + b)x - af(x) for all x.
3.  The quadrilateral ABCD has perpendicular diagonals. Squares ABEF, BCGH, CDIJ and DAKL are constructed on the outside of the sides. The lines CL and DF meet at P, the lines DF and AH meet at Q, the lines AH and BJ meet at R, and the lines BJ and CL meet at S. The lines AI and BK meet at P', the lines BK and CE meet at Q', the lines CE and DG meet at R', and the lines DG and AI meet at S'. Show that the quadrilaterals PQRS and P'Q'R'S' are congruent.
5.  A convex quadrilateral has equal diagonals. Equilateral triangles are constructed on the outside of each side. The centers of the triangles on opposite sides are joined. Show that the two joining lines are perpendicular.
7.  Two circles touch externally at I. The two circles lie inside a large circle and both touch it. The chord BC of the large circle touches both smaller circles (not at I). The common tangent to the two smaller circles at I meets the large circle at A, where A and I are on the same side of the chord BC. Show that I is the incenter of ABC.
8.  Show that there is a convex 1992-gon with sides 1, 2, 3, ... , 1992 in some order, and an inscribed circle (touching every side).
9.  p(x) is a polyonomial with rational coefficients such that k3 - k = 331992 = p(k)3 - p(k) for some real number k. Let pn(x) be p(p( ... p(x) ... )) (iterated n times). Show that pn(k)3 - pn(k) = 331992.
11.  BD and CE are angle bisectors of the triangle ABC. ∠BDE = 24o and ∠CED = 18o. Find angles A, B, C.
12.  The polynomials f(x), g(x) and a(x, y) have real coefficients. They satisfy f(x) - f(y) = a(x, y) ( g(x) - g(y) ) for all x, y. Show that there is a polynomial h(x) such that f(x) = h( g(x) ) for all x.
14.  For each positive integer n, let d(n) be the largest odd divisor of n and define f(n) to be n/2 + n/d(n) for n even and 2(n+1)/2 for n odd. Define the sequence a1, a2, a3, ... by a1 = 1, an+1 = f(an). Show that 1992 occurs in the sequence and find the first time it occurs. Does it occur more than once?
15.  Does there exist a set of 1992 positive integers such that each subset has a sum which is a square, cube or higher power?
16.  Show that (5125 - 1)/(525 - 1) is composite.
17.  Let b(n) be the number of 1s in the binary representation of a positive integer n. Show that b(n2) ≤ b(n) (b(n) + 1)/2 with equality for infinitely many positive integers n. Show that there is a sequence of positive integers a1 < a2 < a3 < ... such that b(an2)/b(an) tends to zero.
18.  Let x1 be a real number such that 0 < x1 < 1. Define xn+1 = 1/xn - [1/xn] for xn non-zero and 0 for xn zero. Show that x1 + x2 + ... + xn < F1/F2 + F2/F3 + ... + Fn/Fn+1, where Fn is the Fibonacci sequence: F1 = F2 = 1, Fn+2 = Fn+1 + Fn.
Note: problems 4, 6, 10, 13, 19 and 20 were used in the Olympiad and are not shown here.

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© John Scholes
6 Aug 2002
Last corrected/updated 25 Nov 03